de Jong: Rationally Connected Varieties

Deformation theory

Let be a nonsingular projective variety over
, and let
be a one-dimensional closed subscheme. We have
, the ideal sheaf of , and we assume that is a local complete intersection, or what is equivalent,
is a locally free sheaf of
-modules of rank . For example, this holds if is a nodal curve.

Definition.
The normal bundle of the curve in is

We have that

where
, and
. We say that is the deformation space, and are the obstructions to deformation. In particular, the dimension the Zariski tangent space of the Hilbert scheme at has
, and it has dimension at least
, the Euler characteristic.

Example.
In the case where
, with is a smooth genus curve, and having at worst simple branchings, then there are no obstructions to deformation and

This is a canonical way of understanding how to move branch points on maps to
.

A map of moduli spaces

Let
be a nonconstant morphism with a nonsingular projective variety over
, and let
be a closed subscheme which is a smooth curve of genus , such that the ramification of is simple. In a (formal) neighborhood of
, the spaces
and the
are the same. The map

where is the degree of on , induces on tangent spaces

induced by the right vertical arrow in the diagram

If is contained in the smooth locus of , then the middle vertical map
is surjective, hence also the right vertical map
is also surjective as maps of sheaves. This will not give a surjection on global sections, however one has:

Corollary. If is contained in the smooth locus of , and
is ``sufficiently positive'', then the morphism

analytically locally around
is a projection
.

One argues that the Hilbert scheme is smooth at the point
since one can twist by a small number of points and keep that the vanishes. In particular, the corollary implies that the morphism is surjective.

We are now ready to prove:

Theorem. [G, Harris, Starr]
If
,
, then any rationally connected variety over for a curve has a rational point.

Proof.
Assume that : all fibres of
are reduced.

Step . Take a general complete intersection
; it will be smooth, irreducible, of say genus and degree . The condition implies that is in the smooth locus and
(by Bertini) has at worst simple branching.

Step . Choose a large integer and choose general points
, and rational curves
such that:

;

;

is very ample;

is in general position.

Now let
. The basic property is that
. Moreover,
with colength and assumption (iv) gives that this is ``general''. This gives that the sheaf
on
is sufficiently positive.

Now deform this curve to a simply branched curve, and this gives the result; conclude by the corollary.

Multiple fibres

We must deal with the case when fails. Suppose we have a family of varieties
with fibres at
irreducible of multiplicity
. Since the curve must intersect these fibres transversally, this must be preserved in any deformation, meaning that the ramification index at will be divisible by .

In this case, the problem is:
cannot dominate. Instead, we consider consider the subset
consisting of stable maps
, of genus , of degree , such that all ramification indices above are equal to .

Now we have the additional problems: Which reducible curves are in
? And perhaps is too small? To resolve both problems, enlarge the genus (but not ) by adding loops to : join two points with a good rational curve. This allows you to break off a component even in this case.

Conclusion

This is work with Jason Starr. What will guarantee the existence of a rational point on a variety over a function field in two variables? Is there a geometric condition which would be like rational connectedness in this case? This is too much to hope for, there are many surfaces with a nontrivial Brauer-Severi variety . Maybe there are geometric restrictions on the fibres such that one obtains a rational section.

A good guess for this condition: demand that certain moduli spaces of rational curves on the fibers are themselves rationally connected. For example, Starr and Harris proved that for hypersurfaces of degree in
with
, the moduli spaces of rational curves of fixed degree on are themselves rationally connected.

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